1THE UNCERTAINTY PRINCIPLE (1927) Werner Heisenberg- Nobel Prize in Physics (1932) at age of 31! Mentor: Niels Bohr The valuesof par�cular pairs of observables cannot be determined simultaneously with arbitrarily high precision in quantum mechanics. Examples of pairs of observables that are restricted in this way are momentum and posi�on, and energy and �me; such pairs are referred to ascomplementary. The quan�ta�ve expressions of the Heisenberg uncertainty principle can be derived by combining the de Broglie rela�onp=h/λand the Einstein rela�onE= hvwith proper�es of all waves. The de Broglie wave for a par�cle is made up of a superposi�on of an infinitely large number of waves of the form () () ,sin 2sin 2x x tAvt Axvt ψπλ π κ =− =−whereAis amplitude andκis the reciprocal wavelength. We consider one spa�al dimension for simplicity. The waves that are added together have infinitesimally different wavelengths. This superposi�on of waves produces awave packetshown to the right. 11 4xxκλπ ∆∆=∆∆≥ (1)(a) weakly localized 1 4tvπ ∆ ∆≥ (2)(b) strongly localized where Δxis the extent of the wave packet in space, Δκis the range in reciprocal wavelength, Δvis the range in frequency, and Δtis a measure of the �me required for the packet to pass a given point. The Δ's in these equa�ons are actually standard devia�ons.
2If at a given �me the wave packet extends over a short range ofxvalues, there is a limit to the accuracy with which we can measure the wavelength. If a wave packet is of short dura�on, there is a limit to the accuracy with which we can measure the frequency. Subs�tu�ng the de Broglie rela�on in equa�on(1), 1/λ =px /hfor mo�on in thexdirec�on, then 14xpxhπ∆∆≥2 x xp∆∆≥ ħ=h/2π(hbar) Another form of the uncertainty principle may be derived by subs�tu�ngE=hvin equa�on (2) and ityields 14Ethπ∆ ∆≥2 tE∆ ∆≥